To unlock the Mirror Cup, collect all postcards in the Postcard Rally races in Story mode.

To unlock the Speedway Reversed bonus track, at the main menu, hold L and press X, SQUARE, TRIANGLE, X, TRIANGLE, SQUARE.

Unlock the following characters by performing the corresponding tasks:

CHARACTER - HOW TO UNLOCK
DJ, Snot Rod, Wingo, Boost, or Chick - Defeat them in their Boss race in Story mode.
Doc Hudson - Win the Gran Prix Cup 1
Flo - With the Gran Prix Cup 1
Sheriff - Win the Gran Prix Cup 2

To unlock alternate paint jobs for the following cars, place in 1st, 2nd, or 3rd in Time Trial mode on the corresponding tracks. You will unlock a different paint job for finishing in 1st, 2nd, or 3rd, for a total of three different paint jobs for each car:

CAR - TRACK
Chick - Speedway track
DJ - Radiator Cap Run track
Doc Hudson - Mesa Ravine track
Lightening McQueen - Figure 8 track
Luigi - Diesel Forest track
Mater - Desert Canyon Dash track
Sally Carrera - Radiator Springs track
Sarge - Frank's Field track
Sheriff - Ironhorse Junction track
Snot Rod - Turbine Canyon
Wingo - Interstate track

The first shortcut is found where there is about five or six obstacles. Jump or drive over them.